Chapter 6 Linear Momentum

Momentum  Momentum is defined as the product of mass and velocity.  p = m·v  Momentum is measured in [kg·m/s]  Momentum is a vector quantity  The momentum of a system of particles is the vector SUM of the individual momenta of each particle.

Example  Comparison of a bullet, a cruise ship, and a glacier…  Qualitative Reasoning…  Quantitative Reasoning…

Newton’s 2 nd Law  Write Newton’s 2 nd Law another way F net = ma F net = m(v f – v i /t) F net = (p f – p i )/t F net = Δp/t

Impulse – Momentum Relationship  F av = Δp/t F av ·Δt = Δp F av ·Δt = p f – p i  Impulse = F av ·Δt measured in [Ns]  When a force acts on an object for a particular amount of time, the impulse it imparts is equivalent to the change in momentum of the object.

Examples  A golfer drives a kg ball from an elevated tee, giving the ball a horizontal speed of 40m/s. What is the magnitude of the average force delivered by the club during this time? The contact time is approximately 1 millisecond.

Examples  A 70.0 kg worker jumps stiff-legged from a height of 1 meter onto a concrete floor. What is the magnitude of the force he feels on landing, assuming a sudden stop in 8.0 milliseconds. –Two parts to the problem: the fall to the floor and the stop by the floor

Kinetic Energy  KE = ½ mv 2 KE = p 2 /(2m)

Conservation of Momentum  Conservation of Momentum is a fundamental concept in physics that allows for analysis of many systems.  It is commonly used to analyze collisions.  Conservation of Momentum can only be applied if no external forces act on a system. Internal forces do not change to overall momentum of a system.  In a closed system, the total momentum of the system is conserved.

Conservation of Momentum  In a closed system: p i = p f p 1i + p 2i + p 3i + … = p 1f + p 2f + p 3f + …

Examples  Two masses m 1 = 1.0 kg and m 2 = 2.0 kg, are held on either side of a light compressed spring by a light string joining them. The string is burned (negligible external force) and the masses move apart on a frictionless surface, with m 1 having a velocity of 1.8 m/s. What is the velocity of m 2 ?

More Examples YAY More Examples YAY  A 30 g bullet with speed of 400 m/s strikes a glancing blow to a target brick of mass 1.0kg. The brick breaks into two fragments. The bullet deflects at an angle of 30° above the x- axis with speed of 100 m/s. One piece of the brick, with mass of 0.75 kg, goes off to the right with a speed of 5.0 m/s. Determine the speed and direction of the other piece.  A physics teacher is lowered from a helicopter to the middle of a smooth level frozen lake. She is challenged to make her way off the ice. Walking is out of the question. (Why?) She decides to throw her identical, heavy mittens, which will provide the momentum to get off the ice. Should she throw them together or separately?

Inelastic vs. Elastic Collisions  For an isolated system, momentum is always conserved whether a collision is elastic or inelastic.  In an inelastic collision, KE is not conserved. Inelastic collisions involve deformation or coupling of individual parts. Some energy goes into the deformation, so mechanical energy is not conserved. Example: Train cars joining, car accidents.  In an elastic collision, KE is conserved in addition to momentum. In an elastic collision, there is no deformation. KE is transferred from one object to another. Example: billiard balls

Example – Inelastic Collision  A 1.0 kg ball with a speed of 4.5 m/s strikes a 2.0 kg stationary ball. If the collision is completely inelastic (the balls stick together after the collision) find the final velocity after the collision. How much Kinetic Energy is lost in this collision?

Center of Mass  The center of mass is the point at which all of the mass of an object or system may be considered to be concentrated.  F net = MA cm where M is total mass of system and A cm is acceleration of the center of mass

Calculating Center of Mass

Examples  Three masses – 2.0 kg, 3.0 kg and 6.0 kg are located at positions (3.0,0), (6.0,0) and (-4.0,0) respectively, in meters, from the origin. Find the center of mass.  A dumbbell has a connecting bar of negligible mass. Find the location of the center of mass if m 1 (located at 0.2 meters) and m 2 (located at 0.9 meters) are each 5.0 kg. What if m 1 is 5.0 kg and m 2 is 10.0 kg?

Center of Gravity  Center of Gravity is similar to Center of Mass – it is the point on an object where the force of gravity is considered to be concentrated.  Many times the location of the center of gravity can be determined by symmetry (circles, squares)  For flat irregularly shaped objects, the center of gravity can be found by suspending the shape from two different points and looking for the intersection (see example in text)